comparison test

The comparison test is the idea that a pile of positive things smaller than a finite pile is itself finite, and a pile bigger than an infinite one is itself infinite. You judge an unfamiliar series by sandwiching its terms against those of a series you already understand.

Stated for series of non-negative terms: suppose 0 <= a_n <= b_n for all n (or for all large n). If sum b_n converges, then sum a_n converges; if sum a_n diverges, then sum b_n diverges. The reasoning is that the partial sums of sum a_n are increasing and bounded above by those of the convergent sum b_n, so by the monotone convergence theorem they converge. The divergence half is the contrapositive.

Two honest cautions. First, the inequality direction matters: a smaller series being bounded by a divergent larger one tells you nothing. Second, the test as stated requires non-negative terms; for series with mixed signs you should compare absolute values, which then yields absolute convergence. The art lies in choosing a good comparison series — usually a geometric series or a p-series.